Squares and products around a centre

Idea: Square from a round neighbour

Stop at the nearest round number, then correct.

• Up: $61^2=3600+120+1=3721$
• Down: $47^2=2500-300+9=2209$
• Going down, the middle term flips sign, the last stays positive

Formula: The correction

$(a\pm b{)}^2=a^2\pm 2ab+b^2$

Square the round part, add or take twice the cross product, add the small square.

Formula: Ending in 5

$(10n+5{)}^2=100n(n+1)+25$

Multiply the leading part by the next integer and write $25$ after it.

Example: Square 85

Compute $85^2$.

1. Leading part $8$, next integer $9$
2. $8\times 9\times 100+25=7225$

Formula: Products around a centre

$(c+d)(c-d)=c^2-d^2$

Factors the same distance from a centre: square the centre, take away the distance squared.

Example: Difference of squares

Compute $56\times 64$.

1. Centre $60$, distance $4$
2. $60^2-4^2=3600-16=3584$

Formula: Both just below a round number

$(c-x)(c-y)=c(c-x-y)+xy$

Distances $x$ and $y$ below $c$: the round part, then add the product of the distances.

Quick check: Near 100

Compute $97\times 94$.

Common mistake: Dropping the middle term

$(a-b{)}^2$ is not $a^2-b^2$: $50^2-3^2=2491$ is really $47\times 53$.

Instead A square keeps its middle term: $47^2=2209$. Name the shape before you start.

Idea: With the clock running

Glance at the sum of the factors first.

• Even sum, easy centre: difference of squares
• Both near $100$: the distance form
• A lone square: its nearest round neighbour